ABSTRACT
The existence of the limit of the numerical sequences can be determined by the Cauchy criterion. In reality, the Cauchy criterion is applicable for complete spaces only. The divergent fundamental sequences of rational numbers determine the irrational numbers by Georg Cantor. By the completion theorem, each fundamental sequence of the arbitrary space is convergent. However, its limits can be elements of the extension of the initial space. The Cantor real numbers are the equivalent classes of the fundamental sequences of rational numbers. The proof of the convergence can be based on the Cauchy criterion that uses the fundamentality of the sequences and does not require a priori knowledge of the limit.
